Question: Determine how many solutions exist for the system of equations. ${-12x-2y = 12}$ ${12x+2y = -12}$
Solution: Convert both equations to slope-intercept form: ${-12x-2y = 12}$ $-12x{+12x} - 2y = 12{+12x}$ $-2y = 12+12x$ $y = -6-6x$ ${y = -6x-6}$ ${12x+2y = -12}$ $12x{-12x} + 2y = -12{-12x}$ $2y = -12-12x$ $y = -6-6x$ ${y = -6x-6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -6x-6}$ ${y = -6x-6}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-12x-2y = 12}$ is also a solution of ${12x+2y = -12}$, there are infinitely many solutions.